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In the last lesson we established that atoms possess discrete energy levels, and that transitions between those levels involve photon emission or absorption at specific frequencies hf = E_upper - E_lower. In this final lesson of the course we apply this theoretical framework to what is perhaps its most direct experimental manifestation: line spectra.
When you look at the light emitted or absorbed by a gas, you do not see a smooth continuum of wavelengths. You see discrete lines — narrow, sharp features at very specific wavelengths — on an otherwise featureless background. Each element produces its own characteristic pattern of lines, as unique as a fingerprint. This is the evidence, visible and measurable, for the existence of quantised atomic energy levels.
Line spectra are central to the OCR A-Level Physics A specification (H556), Module 4.5, and the OCR examiners frequently set questions that require you to (a) relate a spectral line's wavelength to an energy-level transition, (b) explain why line spectra are evidence for quantisation, and (c) distinguish emission and absorption spectra. This lesson develops each of these skills thoroughly.
Before we focus on atomic line spectra, it is useful to set them in context. In 1859, Kirchhoff and Bunsen established three basic categories of optical spectra:
flowchart TB
H["Hot dense body<br/>(filament, Sun's photosphere)"] --> C["Continuous spectrum<br/>all wavelengths present"]
HG["Hot thin gas<br/>(discharge tube)"] --> E["Emission line spectrum<br/>bright lines on dark background"]
CG["Cool thin gas<br/>in front of continuous source"] --> AB["Absorption line spectrum<br/>dark lines on continuous background"]
These three Kirchhoff-Bunsen laws established — decades before Bohr — that atoms interact with light in ways that reveal something about their internal structure. What the discrete lines meant remained a mystery until quantum theory explained them.
When an atom is excited (by collisions, by absorbed photons, or by electric discharge), its electron is promoted to a higher energy level. After a brief time (~10⁻⁸ s), the electron drops back down to a lower level, emitting a photon whose energy exactly equals the energy difference:
hf = E_upper - E_lower
Each possible downward transition corresponds to a specific photon wavelength, and these wavelengths appear as bright lines in the emission spectrum. The number of lines, their positions, and their relative intensities are all characteristic of the element.
For hydrogen, the visible emission lines belong to the Balmer series — transitions that end on n = 2. The strongest four are:
| Transition | Wavelength | Colour |
|---|---|---|
n = 3 → 2 (H-α) | 656.3 nm | red |
n = 4 → 2 (H-β) | 486.1 nm | cyan-blue |
n = 5 → 2 (H-γ) | 434.1 nm | violet |
n = 6 → 2 (H-δ) | 410.2 nm | deep violet |
These four lines are easy to see in a simple hydrogen discharge tube with a diffraction grating. They are direct, visible evidence that the hydrogen atom has a discrete set of energy levels and that transitions between them produce sharp emission lines.
For helium, the emission spectrum shows a different but equally characteristic pattern. For sodium, the familiar yellow "sodium D lines" at 589.0 and 589.6 nm dominate. For neon, a rich pattern of red and orange lines gives a neon sign its distinctive colour. Every element is unique.
When white light — containing all wavelengths — passes through a cool gas, atoms in the gas can absorb photons that match their allowed upward transitions. Photons of the right wavelength are absorbed and the atoms jump to an excited state. The excited atoms then re-emit in random directions, so from the original forward direction these wavelengths appear missing — dark lines against the otherwise continuous background.
Crucially, the dark lines in an absorption spectrum appear at exactly the same wavelengths as the bright lines in an emission spectrum of the same element. The transitions are the same; only the direction (upward for absorption, downward for emission) is reversed.
flowchart LR
W["White light<br/>all wavelengths"] --> CG["Cool gas (Na)"]
CG --> AB["Transmitted light<br/>missing at 589.0, 589.6 nm"]
CG -. re-emits .-> S["Radiation in all directions"]
The most famous absorption spectrum is that of the Sun. When Joseph von Fraunhofer mapped the solar spectrum in 1814, he found thousands of dark lines at discrete wavelengths. These lines are caused by absorption in the Sun's cool outer atmosphere, and each corresponds to a specific element. By matching the wavelengths to laboratory spectra, astronomers have identified the chemical composition of the Sun — and, by extension, of every star and galaxy we can observe.
This technique, astronomical spectroscopy, is one of the most powerful tools in astrophysics. The helium atom was first identified in the spectrum of the Sun (helios = Sun) in 1868 — twenty-seven years before it was isolated in a laboratory on Earth. Today, every discovery about the composition, motion, and history of distant objects in the universe ultimately traces back to the same equation: hf = E_upper - E_lower.
The discrete nature of atomic spectra is direct evidence that atomic energy levels are themselves discrete. If the energy levels formed a continuum — any energy allowed — then transitions between them would produce photons of any energy, and the spectrum would be continuous (like the blackbody spectrum). The fact that we see sharp lines instead of a smeared continuum is evidence — overwhelming and unambiguous — that atoms possess only a discrete set of allowed energies.
This is the key point that OCR exam questions often test, and it is worth rehearsing explicitly:
Emission line spectra are evidence for discrete atomic energy levels because each observed line corresponds to a specific photon energy
hf = E_upper - E_lower. Since only certain discrete photon energies are observed (not a continuous range), only certain discrete energy differences — and therefore only certain discrete energy levels — can exist within the atom.
This kind of concise, logically clear explanation is exactly what is expected in a 5- or 6-mark exam question.
Exam Tip: When OCR asks "explain how line spectra provide evidence for discrete energy levels", make sure your answer explicitly includes: (1) each line corresponds to a transition between two specific levels, (2) the photon energy is
hf = E_upper - E_lower, (3) sharp discrete lines (not a continuum) imply discrete energy differences, which in turn imply discrete energy levels. All three steps in the logical chain are needed for full marks.
A hydrogen atom makes a transition from n = 4 to n = 2. What is the wavelength of the emitted photon? Which of the named Balmer lines is this?
Solution.
E_4 = -13.6/4² = -0.85 eV; E_2 = -13.6/4 = -3.40 eV.
ΔE = E_4 - E_2 = -0.85 - (-3.40) = 2.55 eV
(Writing E_upper - E_lower as always — a positive number.)
Wavelength:
λ = 1240/2.55 ≈ 486 nm
Answer: 486 nm — this is the H-β line of the Balmer series, in the cyan-blue part of the visible spectrum.
A beam of white light passes through a cloud of cold hydrogen atoms in their ground state. Which wavelengths are absorbed? Which are not?
Solution. From the ground state (n = 1, E = -13.6 eV), the available upward transitions are to n = 2, n = 3, n = 4, ... Each requires a specific photon energy:
1 → 2: ΔE = 10.20 eV, λ = 121.6 nm (Lyman-α, UV)1 → 3: ΔE = 12.09 eV, λ = 102.6 nm (Lyman-β, UV)1 → 4: ΔE = 12.75 eV, λ = 97.3 nm (Lyman-γ, UV)1 → ∞: ΔE = 13.60 eV, λ = 91.2 nm (Lyman limit, UV)Notice that all Lyman absorption lines are in the ultraviolet. Ground-state hydrogen does not absorb visible light — this is why pure hydrogen gas is transparent and colourless. The visible Balmer lines (red, blue, violet) would require the atom to start in the n = 2 excited state, which is only populated at very high temperatures or in active discharge tubes. Cold hydrogen in space shows Lyman absorption in the UV but is invisible in visible wavelengths.
Answer: Only UV wavelengths at 121.6, 102.6, 97.3, ... nm are absorbed. Visible light passes through unchanged.
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